A328722 Dirichlet g.f.: 1 / zeta(s-1)^2.

1, -4, -6, 4, -10, 24, -14, 0, 9, 40, -22, -24, -26, 56, 60, 0, -34, -36, -38, -40, 84, 88, -46, 0, 25, 104, 0, -56, -58, -240, -62, 0, 132, 136, 140, 36, -74, 152, 156, 0, -82, -336, -86, -88, -90, 184, -94, 0, 49, -100, 204, -104, -106, 0, 220, 0, 228, 232, -118, 240

Offset: 1

Author

Ilya Gutkovskiy, Oct 26 2019

Keywords

• sign
• easy
• mult

Comments

Dirichlet inverse of A038040.

Dirichlet convolution of A055615 with itself.

Moebius transform of A101035.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A007427, A008683, A038040, A046099 (positions of 0's), A055615, A101035.

Programs

Mathematica

a[1] = 1; a[n_] := -Sum[(n/d) DivisorSigma[0, n/d] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 60}]
Table[n DivisorSum[n, MoebiusMu[n/#] MoebiusMu[#] &], {n, 1, 60}]
f[p_, e_] := Switch[e, 1, -2*p, 2, p^2, , 0]; a[1] = 1; a[n] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2023 *)

PARI

for(n=1, 100, print1(direuler(p=2, n, (1 - p*X)^2)[n], ", ")) \\ Vaclav Kotesovec, Aug 22 2021

PARI

A007427(n) = if(n<1, 0, sumdiv(n, d, moebius(d) * moebius(n/d))); \\ From A007427
A328722(n) = (n*A007427(n)); \\ Antti Karttunen, Nov 15 2021

Formula

a(1) = 1; a(n) = -Sum_{d|n, dA038040(n/d) * a(d).

a(n) = n * A007427(n).

a(n) = Sum_{d|n} mu(n/d) * A101035(d).

Multiplicative with a(p) = -2*p, a(p^2) = p^2, and a(p) = 0 for e >= 3. - Amiram Eldar, Sep 15 2023